Velocity Reconstruction from Flow-Induced Magnetic Fields

TL;DR

This paper investigates the inverse problem of reconstructing an incompressible velocity field from magnetic field observations, analyzing conditions for uniqueness and well-posedness in different domain settings.

Contribution

It provides new criteria for the uniqueness and solvability of velocity reconstruction from magnetic data, including Diophantine conditions and domain-dependent criteria.

Findings

Reconstruction is well-posed on non-characteristic hypersurfaces in  space.

Sharp uniqueness criterion on a torus depends on rational dependence of background field ratios.

Velocity reconstruction in L^2 requires the background field to satisfy a Diophantine condition.

Abstract

We study the inverse problem of reconstructing an incompressible velocity field $\boldsymbol{v}$ from observations of the induced magnetic field $\boldsymbol{b}$. In the presence of a strong, constant background field $\mathbf{F}$, the evolution of the magnetic perturbation $\boldsymbol{b}$ is governed by the linearized induction equation. We analyze the system on both the entire space $\Omega = \mathbb{R}^d$ and a periodic domain $\Omega = \prod_{i=1}^d [0, L_i)$, which models a homogeneous medium with side lengths $L_i > 0$. We analyze this problem by decomposing it into the injectivity of a parabolic forward map and the solvability of a divergence-free transport sub-problem. On the whole space $\mathbb{R}^d$, we show that the transport sub-problem is well-posed when data is prescribed on a non-characteristic hypersurface transverse to $\mathbf{F}$. On the torus, we establish a sharp uniqueness criterion based on the rational dependence of the ratios $\{F_i/L_i\}_{i=1}^d$ between the background-field components and the corresponding domain periods. Furthermore, we show that for the reconstructed velocity to belong to $L^2$, a sufficient condition is that the background field must satisfy a Diophantine condition. The proof combines injectivity of the parabolic forward map with uniqueness for a steady transport equation along $\mathbf{F}$.